# LLN and CLT¶

Contents

## Overview¶

This lecture illustrates two of the most important theorems of probability and statistics: The law of large numbers (LLN) and the central limit theorem (CLT)

These beautiful theorems lie behind many of the most fundamental results in econometrics and quantitative economic modeling

The lecture is based around simulations that show the LLN and CLT in action

We also demonstrate how the LLN and CLT break down when the assumptions they are based on do not hold

In addition, we examine several useful extensions of the classical theorems, such as

• The delta method, for smooth functions of random variables
• The multivariate case

Some of these extensions are presented as exercises

## Relationships¶

The CLT refines the LLN

The LLN gives conditions under which sample moments converge to population moments as sample size increases

The CLT provides information about the rate at which sample moments converge to population moments as sample size increases

## LLN¶

We begin with the law of large numbers, which tells us when sample averages will converge to their population means

### The Classical LLN¶

The classical law of large numbers concerns independent and identically distributed (IID) random variables

Here is the strongest version of the classical LLN, known as Kolmogorov’s strong law

Let $$X_1, \ldots, X_n$$ be independent and identically distributed scalar random variables, with common distribution $$F$$

When it exists, let $$\mu$$ denote the common mean of this sample:

$\mu := \mathbb E X = \int x F(dx)$

$\bar X_n := \frac{1}{n} \sum_{i=1}^n X_i$

Kolmogorov’s strong law states that, if $$\mathbb E |X|$$ is finite, then

(1)$\mathbb P \left\{ \bar X_n \to \mu \text{ as } n \to \infty \right\} = 1$

What does this last expression mean?

Let’s think about it from a simulation perspective, imagining for a moment that our computer can generate perfect random samples (which of course it can’t)

Let’s also imagine that we can generate infinite sequences, so that the statement $$\bar X_n \to \mu$$ can be evaluated

In this setting, (1) should be interpreted as meaning that the probability of the computer producing a sequence where $$\bar X_n \to \mu$$ fails to occur is zero

### Proof¶

The proof of Kolmogorov’s strong law is nontrivial – see, for example, theorem 8.3.5 of [Dud02]

On the other hand, we can prove a weaker version of the LLN very easily and still get most of the intuition

The version we prove is as follows: If $$X_1, \ldots, X_n$$ is IID with $$\mathbb E X_i^2 < \infty$$, then, for any $$\epsilon > 0$$, we have

(2)$\mathbb P \left\{ | \bar X_n - \mu | \geq \epsilon \right\} \to 0 \quad \text{as} \quad n \to \infty$

(This version is weaker because we claim only convergence in probability rather than almost sure convergence, and assume a finite second moment)

To see that this is so, fix $$\epsilon > 0$$, and let $$\sigma^2$$ be the variance of each $$X_i$$

Recall the Chebyshev inequality, which tells us that

(3)$\mathbb P \left\{ | \bar X_n - \mu | \geq \epsilon \right\} \leq \frac{\mathbb E [ (\bar X_n - \mu)^2]}{\epsilon^2}$

Now observe that

\begin{split}\begin{aligned} \mathbb E [ (\bar X_n - \mu)^2 ] & = \mathbb E \left\{ \left[ \frac{1}{n} \sum_{i=1}^n (X_i - \mu) \right]^2 \right\} \\ & = \frac{1}{n^2} \sum_{i=1}^n \sum_{j=1}^n \mathbb E (X_i - \mu)(X_j - \mu) \nonumber \\ & = \frac{1}{n^2} \sum_{i=1}^n \mathbb E (X_i - \mu)^2 \nonumber \\ & = \frac{\sigma^2}{n} \nonumber \end{aligned}\end{split}

Here the crucial step is at the third equality, which follows from independence

Independence means that if $$i \not= j$$, then the covariance term $$\mathbb E (X_i - \mu)(X_j - \mu)$$ drops out

As a result, $$n^2 - n$$ terms vanish, leading us to a final expression that goes to zero in $$n$$

Combining our last result with (3), we come to the estimate

(4)$\mathbb P \left\{ | \bar X_n - \mu | \geq \epsilon \right\} \leq \frac{\sigma^2}{n \epsilon^2}$

The claim in (2) is now clear

Of course, if the sequence $$X_1, \ldots, X_n$$ is correlated, then the cross-product terms $$\mathbb E (X_i - \mu)(X_j - \mu)$$ are not necessarily zero

While this doesn’t mean that the same line of argument is impossible, it does mean that if we want a similar result then the covariances should be “almost zero” for “most” of these terms

In a long sequence, this would be true if, for example, $$\mathbb E (X_i - \mu)(X_j - \mu)$$ approached zero when the difference between $$i$$ and $$j$$ became large

In other words, the LLN can still work if the sequence $$X_1, \ldots, X_n$$ has a kind of “asymptotic independence”, in the sense that correlation falls to zero as variables become further apart in the sequence

This idea is very important in time series analysis, and we’ll come across it again soon enough

### Illustration¶

Let’s now illustrate the classical IID law of large numbers using simulation

In particular, we aim to generate some sequences of IID random variables and plot the evolution of $$\bar X_n$$ as $$n$$ increases

Below is a figure that does just this (as usual, you can click on it to expand it)

It shows IID observations from three different distributions and plots $$\bar X_n$$ against $$n$$ in each case

The dots represent the underlying observations $$X_i$$ for $$i = 1, \ldots, 100$$

In each of the three cases, convergence of $$\bar X_n$$ to $$\mu$$ occurs as predicted

The figure was produced by file illustrates_lln.jl, and can be downloaded here

The three distributions are chosen at random from a selection stored in the dictionary distributions

#=
Visual illustration of the law of large numbers.

@author : Spencer Lyon <spencer.lyon@nyu.edu>
Victoria Gregory <victoria.gregory@nyu.edu>

References
----------

Based off the original python file illustrates_lln.py
=#
using Plots
pyplot()
using Distributions
using LaTeXStrings

n = 100
srand(42)  # reproducible results

# == Arbitrary collection of distributions == #
distributions = Dict("student's t with 10 degrees of freedom" => TDist(10),
"beta(2, 2)" => Beta(2.0, 2.0),
"lognormal LN(0, 1/2)" => LogNormal(0.5),
"gamma(5, 1/2)" => Gamma(5.0, 2.0),
"poisson(4)" => Poisson(4),
"exponential with lambda = 1" => Exponential(1))

num_plots = 3
dist_data = zeros(num_plots, n)
sample_means = []
dist_means = []
titles = []
for i = 1:num_plots
dist_names = collect(keys(distributions))
# == Choose a randomly selected distribution == #
name = dist_names[rand(1:length(dist_names))]
dist = pop!(distributions, name)

# == Generate n draws from the distribution == #
data = rand(dist, n)

# == Compute sample mean at each n == #
sample_mean = Array(Float64, n)
for j=1:n
sample_mean[j] = mean(data[1:j])
end

m = mean(dist)

dist_data[i, :] = data'
push!(sample_means, sample_mean)
push!(dist_means, m*ones(n))
push!(titles, name)

end

# == Plot == #
N = repmat(reshape(repmat(linspace(1, n, n), 1, num_plots)', 1, n*num_plots), 2, 1)
heights = [zeros(1,n*num_plots); reshape(dist_data, 1, n*num_plots)]
plot(N, heights, layout=(3, 1), label="", color=:grey, alpha=0.5)
plot!(1:n, dist_data', layout=(3, 1), color=:grey, markershape=:circle,
alpha=0.5, label="", linewidth=0)
plot!(1:n, sample_means, linewidth=3, alpha=0.6, color=:green, legend=:topleft,
layout=(3, 1), label=[LaTeXString("\$\\bar{X}_n\$") "" ""])
plot!(1:n, dist_means, color=:black, linewidth=1.5, layout=(3, 1),
linestyle=:dash, grid=false, label=[LaTeXString("\$\\mu\$") "" ""])
plot!(title=titles')


### Infinite Mean¶

What happens if the condition $$\mathbb E | X | < \infty$$ in the statement of the LLN is not satisfied?

This might be the case if the underlying distribution is heavy tailed — the best known example is the Cauchy distribution, which has density

$f(x) = \frac{1}{\pi (1 + x^2)} \qquad (x \in \mathbb R)$

The next figure shows 100 independent draws from this distribution

Notice how extreme observations are far more prevalent here than the previous figure

Let’s now have a look at the behavior of the sample mean

Here we’ve increased $$n$$ to 1000, but the sequence still shows no sign of converging

Will convergence become visible if we take $$n$$ even larger?

To see this, recall that the characteristic function of the Cauchy distribution is

(5)$\phi(t) = \mathbb E e^{itX} = \int e^{i t x} f(x) dx = e^{-|t|}$

Using independence, the characteristic function of the sample mean becomes

\begin{split}\begin{aligned} \mathbb E e^{i t \bar X_n } & = \mathbb E \exp \left\{ i \frac{t}{n} \sum_{j=1}^n X_j \right\} \\ & = \mathbb E \prod_{j=1}^n \exp \left\{ i \frac{t}{n} X_j \right\} \\ & = \prod_{j=1}^n \mathbb E \exp \left\{ i \frac{t}{n} X_j \right\} = [\phi(t/n)]^n \end{aligned}\end{split}

In view of (5), this is just $$e^{-|t|}$$

Thus, in the case of the Cauchy distribution, the sample mean itself has the very same Cauchy distribution, regardless of $$n$$

In particular, the sequence $$\bar X_n$$ does not converge to a point

## CLT¶

Next we turn to the central limit theorem, which tells us about the distribution of the deviation between sample averages and population means

### Statement of the Theorem¶

The central limit theorem is one of the most remarkable results in all of mathematics

In the classical IID setting, it tells us the following:

If the sequence $$X_1, \ldots, X_n$$ is IID, with common mean $$\mu$$ and common variance $$\sigma^2 \in (0, \infty)$$, then

(6)$\sqrt{n} ( \bar X_n - \mu ) \stackrel { d } {\to} N(0, \sigma^2) \quad \text{as} \quad n \to \infty$

Here $$\stackrel { d } {\to} N(0, \sigma^2)$$ indicates convergence in distribution to a centered (i.e, zero mean) normal with standard deviation $$\sigma$$

### Intuition¶

The striking implication of the CLT is that for any distribution with finite second moment, the simple operation of adding independent copies always leads to a Gaussian curve

A relatively simple proof of the central limit theorem can be obtained by working with characteristic functions (see, e.g., theorem 9.5.6 of [Dud02])

The proof is elegant but almost anticlimactic, and it provides surprisingly little intuition

In fact all of the proofs of the CLT that we know are similar in this respect

Why does adding independent copies produce a bell-shaped distribution?

Part of the answer can be obtained by investigating addition of independent Bernoulli random variables

In particular, let $$X_i$$ be binary, with $$\mathbb P\{X_i = 0\} = \mathbb P\{X_i = 1 \} = 0.5$$, and let $$X_1, \ldots, X_n$$ be independent

Think of $$X_i = 1$$ as a “success”, so that $$Y_n = \sum_{i=1}^n X_i$$ is the number of successes in $$n$$ trials

The next figure plots the probability mass function of $$Y_n$$ for $$n = 1, 2, 4, 8$$

When $$n = 1$$, the distribution is flat — one success or no successes have the same probability

When $$n = 2$$ we can either have 0, 1 or 2 successes

Notice the peak in probability mass at the mid-point $$k=1$$

The reason is that there are more ways to get 1 success (“fail then succeed” or “succeed then fail”) than to get zero or two successes

Moreover, the two trials are independent, so the outcomes “fail then succeed” and “succeed then fail” are just as likely as the outcomes “fail then fail” and “succeed then succeed”

(If there was positive correlation, say, then “succeed then fail” would be less likely than “succeed then succeed”)

Here, already we have the essence of the CLT: addition under independence leads probability mass to pile up in the middle and thin out at the tails

For $$n = 4$$ and $$n = 8$$ we again get a peak at the “middle” value (halfway between the minimum and the maximum possible value)

The intuition is the same — there are simply more ways to get these middle outcomes

If we continue, the bell-shaped curve becomes ever more pronounced

We are witnessing the binomial approximation of the normal distribution

### Simulation 1¶

Since the CLT seems almost magical, running simulations that verify its implications is one good way to build intuition

To this end, we now perform the following simulation

1. Choose an arbitrary distribution $$F$$ for the underlying observations $$X_i$$
2. Generate independent draws of $$Y_n := \sqrt{n} ( \bar X_n - \mu )$$
3. Use these draws to compute some measure of their distribution — such as a histogram
4. Compare the latter to $$N(0, \sigma^2)$$

Here’s some code that does exactly this for the exponential distribution $$F(x) = 1 - e^{- \lambda x}$$

(Please experiment with other choices of $$F$$, but remember that, to conform with the conditions of the CLT, the distribution must have finite second moment)

#=
Visual illustration of the central limit theorem

@author : Spencer Lyon <spencer.lyon@nyu.edu>
Victoria Gregory <victoria.gregory@nyu.edu>

References
----------

Based off the original python file illustrates_clt.py
=#
using Plots
pyplot()
using Distributions
using LaTeXStrings

# == Set parameters == #
srand(42)  # reproducible results
n = 250    # Choice of n
k = 100000  # Number of draws of Y_n
dist = Exponential(1./2.)  # Exponential distribution, lambda = 1/2
mu, s = mean(dist), std(dist)

# == Draw underlying RVs. Each row contains a draw of X_1,..,X_n == #
data = rand(dist, (k, n))

# == Compute mean of each row, producing k draws of \bar X_n == #
sample_means = mean(data, 2)

# == Generate observations of Y_n == #
Y = sqrt(n) * (sample_means .- mu)

# == Plot == #
xmin, xmax = -3 * s, 3 * s
histogram(Y, nbins=60, alpha=0.5, xlims=(xmin, xmax),
norm=true, label="")
xgrid = linspace(xmin, xmax, 200)
plot!(xgrid, pdf(Normal(0.0, s), xgrid), color=:black,
linewidth=2, label=LaTeXString("\$N(0, \\sigma^2=$(s^2))\\$"),
legendfont=font(12))


The file is illustrates_clt.jl, and can be downloaded here

The program produces figures such as the one below

The fit to the normal density is already tight, and can be further improved by increasing n

You can also experiment with other specifications of $$F$$

### Simulation 2¶

Our next simulation is somewhat like the first, except that we aim to track the distribution of $$Y_n := \sqrt{n} ( \bar X_n - \mu )$$ as $$n$$ increases

In the simulation we’ll be working with random variables having $$\mu = 0$$

Thus, when $$n=1$$, we have $$Y_1 = X_1$$, so the first distribution is just the distribution of the underlying random variable

For $$n=2$$, the distribution of $$Y_2$$ is that of $$(X_1 + X_2) / \sqrt{2}$$, and so on

What we expect is that, regardless of the distribution of the underlying random variable, the distribution of $$Y_n$$ will smooth out into a bell shaped curve

The next figure shows this process for $$X_i \sim f$$, where $$f$$ was specified as the convex combination of three different beta densities

(Taking a convex combination is an easy way to produce an irregular shape for $$f$$)

In the figure, the closest density is that of $$Y_1$$, while the furthest is that of $$Y_5$$

As expected, the distribution smooths out into a bell curve as $$n$$ increases

The figure is generated by file clt3d.jl, and can be downloaded here

We leave you to investigate its contents if you wish to know more

If you run the file from the ordinary Julia or IJulia shell, the figure should pop up in a window that you can rotate with your mouse, giving different views on the density sequence

### The Multivariate Case¶

The law of large numbers and central limit theorem work just as nicely in multidimensional settings

To state the results, let’s recall some elementary facts about random vectors

A random vector $$\mathbf X$$ is just a sequence of $$k$$ random variables $$(X_1, \ldots, X_k)$$

Each realization of $$\mathbf X$$ is an element of $$\mathbb R^k$$

A collection of random vectors $$\mathbf X_1, \ldots, \mathbf X_n$$ is called independent if, given any $$n$$ vectors $$\mathbf x_1, \ldots, \mathbf x_n$$ in $$\mathbb R^k$$, we have

$\mathbb P\{\mathbf X_1 \leq \mathbf x_1,\ldots, \mathbf X_n \leq \mathbf x_n \} = \mathbb P\{\mathbf X_1 \leq \mathbf x_1 \} \times \cdots \times \mathbb P\{ \mathbf X_n \leq \mathbf x_n \}$

(The vector inequality $$\mathbf X \leq \mathbf x$$ means that $$X_j \leq x_j$$ for $$j = 1,\ldots,k$$)

Let $$\mu_j := \mathbb E [X_j]$$ for all $$j =1,\ldots,k$$

The expectation $$\mathbb E [\mathbf X]$$ of $$\mathbf X$$ is defined to be the vector of expectations:

$\begin{split}\mathbb E [\mathbf X] := \left( \begin{array}{c} \mathbb E [X_1] \\ \mathbb E [X_2] \\ \vdots \\ \mathbb E [X_k] \end{array} \right) = \left( \begin{array}{c} \mu_1 \\ \mu_2\\ \vdots \\ \mu_k \end{array} \right) =: \boldsymbol \mu\end{split}$

The variance-covariance matrix of random vector $$\mathbf X$$ is defined as

$\Var[\mathbf X] := \mathbb E [ (\mathbf X - \boldsymbol \mu) (\mathbf X - \boldsymbol \mu)']$

Expanding this out, we get

$\begin{split}\Var[\mathbf X] = \left( \begin{array}{ccc} \mathbb E [(X_1 - \mu_1)(X_1 - \mu_1)] & \cdots & \mathbb E [(X_1 - \mu_1)(X_k - \mu_k)] \\ \mathbb E [(X_2 - \mu_2)(X_1 - \mu_1)] & \cdots & \mathbb E [(X_2 - \mu_2)(X_k - \mu_k)] \\ \vdots & \vdots & \vdots \\ \mathbb E [(X_k - \mu_k)(X_1 - \mu_1)] & \cdots & \mathbb E [(X_k - \mu_k)(X_k - \mu_k)] \\ \end{array} \right)\end{split}$

The $$j,k$$-th term is the scalar covariance between $$X_j$$ and $$X_k$$

With this notation we can proceed to the multivariate LLN and CLT

Let $$\mathbf X_1, \ldots, \mathbf X_n$$ be a sequence of independent and identically distributed random vectors, each one taking values in $$\mathbb R^k$$

Let $$\boldsymbol \mu$$ be the vector $$\mathbb E [\mathbf X_i]$$, and let $$\Sigma$$ be the variance-covariance matrix of $$\mathbf X_i$$

Interpreting vector addition and scalar multiplication in the usual way (i.e., pointwise), let

$\bar{\mathbf X}_n := \frac{1}{n} \sum_{i=1}^n \mathbf X_i$

In this setting, the LLN tells us that

(7)$\mathbb P \left\{ \bar{\mathbf X}_n \to \boldsymbol \mu \text{ as } n \to \infty \right\} = 1$

Here $$\bar{\mathbf X}_n \to \boldsymbol \mu$$ means that $$\| \bar{\mathbf X}_n \to \boldsymbol \mu \| \to 0$$, where $$\| \cdot \|$$ is the standard Euclidean norm

The CLT tells us that, provided $$\Sigma$$ is finite,

(8)$\sqrt{n} ( \bar{\mathbf X}_n - \boldsymbol \mu ) \stackrel { d } {\to} N(\mathbf 0, \Sigma) \quad \text{as} \quad n \to \infty$

## Exercises¶

### Exercise 1¶

One very useful consequence of the central limit theorem is as follows

Assume the conditions of the CLT as stated above

If $$g \colon \mathbb R \to \mathbb R$$ is differentiable at $$\mu$$ and $$g'(\mu) \not= 0$$, then

(9)$\sqrt{n} \{ g(\bar X_n) - g(\mu) \} \stackrel { d } {\to} N(0, g'(\mu)^2 \sigma^2) \quad \text{as} \quad n \to \infty$

This theorem is used frequently in statistics to obtain the asymptotic distribution of estimators — many of which can be expressed as functions of sample means

(These kinds of results are often said to use the “delta method”)

The proof is based on a Taylor expansion of $$g$$ around the point $$\mu$$

Taking the result as given, let the distribution $$F$$ of each $$X_i$$ be uniform on $$[0, \pi / 2]$$ and let $$g(x) = \sin(x)$$

Derive the asymptotic distribution of $$\sqrt{n} \{ g(\bar X_n) - g(\mu) \}$$ and illustrate convergence in the same spirit as the program illustrate_clt.jl discussed above

What happens when you replace $$[0, \pi / 2]$$ with $$[0, \pi]$$?

What is the source of the problem?

### Exercise 2¶

Here’s a result that’s often used in developing statistical tests, and is connected to the multivariate central limit theorem

If you study econometric theory, you will see this result used again and again

Assume the setting of the multivariate CLT discussed above, so that

1. $$\mathbf X_1, \ldots, \mathbf X_n$$ is a sequence of IID random vectors, each taking values in $$\mathbb R^k$$
2. $$\boldsymbol \mu := \mathbb E [\mathbf X_i]$$, and $$\Sigma$$ is the variance-covariance matrix of $$\mathbf X_i$$
3. The convergence
(10)$\sqrt{n} ( \bar{\mathbf X}_n - \boldsymbol \mu ) \stackrel { d } {\to} N(\mathbf 0, \Sigma)$

is valid

In a statistical setting, one often wants the right hand side to be standard normal, so that confidence intervals are easily computed

This normalization can be achieved on the basis of three observations

First, if $$\mathbf X$$ is a random vector in $$\mathbb R^k$$ and $$\mathbf A$$ is constant and $$k \times k$$, then

$\Var[\mathbf A \mathbf X] = \mathbf A \Var[\mathbf X] \mathbf A'$

Second, by the continuous mapping theorem, if $$\mathbf Z_n \stackrel{d}{\to} \mathbf Z$$ in $$\mathbb R^k$$ and $$\mathbf A$$ is constant and $$k \times k$$, then

$\mathbf A \mathbf Z_n \stackrel{d}{\to} \mathbf A \mathbf Z$

Third, if $$\mathbf S$$ is a $$k \times k$$ symmetric positive definite matrix, then there exists a symmetric positive definite matrix $$\mathbf Q$$, called the inverse square root of $$\mathbf S$$, such that

$\mathbf Q \mathbf S\mathbf Q' = \mathbf I$

Here $$\mathbf I$$ is the $$k \times k$$ identity matrix

Putting these things together, your first exercise is to show that if $$\mathbf Q$$ is the inverse square root of $$\mathbf \Sigma$$, then

$\mathbf Z_n := \sqrt{n} \mathbf Q ( \bar{\mathbf X}_n - \boldsymbol \mu ) \stackrel{d}{\to} \mathbf Z \sim N(\mathbf 0, \mathbf I)$

Applying the continuous mapping theorem one more time tells us that

$\| \mathbf Z_n \|^2 \stackrel{d}{\to} \| \mathbf Z \|^2$

Given the distribution of $$\mathbf Z$$, we conclude that

(11)$n \| \mathbf Q ( \bar{\mathbf X}_n - \boldsymbol \mu ) \|^2 \stackrel{d}{\to} \chi^2(k)$

where $$\chi^2(k)$$ is the chi-squared distribution with $$k$$ degrees of freedom

(Recall that $$k$$ is the dimension of $$\mathbf X_i$$, the underlying random vectors)

Your second exercise is to illustrate the convergence in (11) with a simulation

In doing so, let

$\begin{split}\mathbf X_i := \left( \begin{array}{c} W_i \\ U_i + W_i \end{array} \right)\end{split}$

where

• each $$W_i$$ is an IID draw from the uniform distribution on $$[-1, 1]$$
• each $$U_i$$ is an IID draw from the uniform distribution on $$[-2, 2]$$
• $$U_i$$ and $$W_i$$ are independent of each other

Hints:

1. sqrtm(A) computes the square root of A. You still need to invert it
2. You should be able to work out $$\Sigma$$ from the proceding information

## Solutions¶

Solution notebook

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